| L(s) = 1 | + 2-s + 1.70·3-s + 4-s − 3.85·5-s + 1.70·6-s + 1.93·7-s + 8-s − 0.0938·9-s − 3.85·10-s − 0.959·11-s + 1.70·12-s − 4.27·13-s + 1.93·14-s − 6.56·15-s + 16-s − 0.472·17-s − 0.0938·18-s + 2.27·19-s − 3.85·20-s + 3.30·21-s − 0.959·22-s − 0.615·23-s + 1.70·24-s + 9.83·25-s − 4.27·26-s − 5.27·27-s + 1.93·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.984·3-s + 0.5·4-s − 1.72·5-s + 0.695·6-s + 0.731·7-s + 0.353·8-s − 0.0312·9-s − 1.21·10-s − 0.289·11-s + 0.492·12-s − 1.18·13-s + 0.517·14-s − 1.69·15-s + 0.250·16-s − 0.114·17-s − 0.0221·18-s + 0.521·19-s − 0.861·20-s + 0.720·21-s − 0.204·22-s − 0.128·23-s + 0.347·24-s + 1.96·25-s − 0.838·26-s − 1.01·27-s + 0.365·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 - 1.70T + 3T^{2} \) |
| 5 | \( 1 + 3.85T + 5T^{2} \) |
| 7 | \( 1 - 1.93T + 7T^{2} \) |
| 11 | \( 1 + 0.959T + 11T^{2} \) |
| 13 | \( 1 + 4.27T + 13T^{2} \) |
| 17 | \( 1 + 0.472T + 17T^{2} \) |
| 19 | \( 1 - 2.27T + 19T^{2} \) |
| 23 | \( 1 + 0.615T + 23T^{2} \) |
| 29 | \( 1 - 6.02T + 29T^{2} \) |
| 31 | \( 1 + 5.98T + 31T^{2} \) |
| 37 | \( 1 + 0.683T + 37T^{2} \) |
| 41 | \( 1 - 1.35T + 41T^{2} \) |
| 43 | \( 1 - 0.497T + 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 - 3.02T + 53T^{2} \) |
| 59 | \( 1 + 14.8T + 59T^{2} \) |
| 61 | \( 1 + 2.16T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 9.61T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 6.16T + 79T^{2} \) |
| 83 | \( 1 + 9.66T + 83T^{2} \) |
| 89 | \( 1 + 7.83T + 89T^{2} \) |
| 97 | \( 1 + 8.80T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.906758631442850862073101203704, −7.56916320022133834902906662036, −6.90594994389750731159215271737, −5.65532578499692485928658818078, −4.68393707163885247515284899487, −4.37453066724780136151030614026, −3.24048245508644708717640319996, −2.94117500917561426397053498060, −1.72808844101672235651085291992, 0,
1.72808844101672235651085291992, 2.94117500917561426397053498060, 3.24048245508644708717640319996, 4.37453066724780136151030614026, 4.68393707163885247515284899487, 5.65532578499692485928658818078, 6.90594994389750731159215271737, 7.56916320022133834902906662036, 7.906758631442850862073101203704
